The n-Category Café

The often mocked, vastly underappreciated mathematician and astronomer Claudius Ptolemy proved in his Almagest that:

A quadrilateral can be inscribed in a circle whenever the product of the lengths of the diagonals equals the sum of the products of the lengths of opposite sides.

In modern notation, four points $A,B,C,D \in \mathbb{R}^2$ lie on a circle if they satisfy this equation:

$$\| A - C \| \; \| B - D\| = \|A - B \| \; \|C - D \| \; + \; \|A - D\| \; \| B - C \|$$

On the other hand, 4 points $A,B,C,D \in \mathbb{R}$ always satisfy this strikingly similar equation:

$$(A - C) \; ( B - D) = (A - B)\;(C - D ) \; + \; (A - D)\;(B - C)$$

A related equation shows up in yet another context: the theory of flows. But here, ‘max’ plays the role of addition and ‘$+$’ plays the role of multiplication. What’s going on? Gavin explains all.

But now Gavin Wraith has a question that requires help! — help from someone who knows elliptic curves and old-fashioned synthetic geometry. It’s about Poncelet’s Porism.

First of all, what the heck is a ‘porism’?

This is one of those scary Greek math words like ‘syzygy’ and ‘plethysm’ — words that nobody ever seems to explain in a clear, intuitive way. Click on the links and you’ll see what I mean! I’ll explain ‘em someday… but not today.

It’s not promising that the Wikipedia entry for ‘porism’ begins:

The subject of porisms is perplexed by the multitude of different views which have been held by geometers as to what a porism really was and is.

In brief, a porism is something in between a ‘problem’ and a ‘theorem’. Perhaps this explanation is as good as it gets:

The older geometers regarded a theorem as directed to proving what is proposed, a problem as directed to constructing what is proposed, and finally a porism as directed to finding what is proposed.

But never mind. Poncelet’s porism goes like this:

Let $C$ and $D$ be two plane conics. If you can find, one $n$-sided polygon ($n \ge 3$) that’s inscribed in $C$ and circumscribed around $D$, then you can find infinitely many of them.

A modern treatment quickly gets into elliptic curves, but I don’t understand it yet. Anyway, here’s Gavin’s question:

Dear John -

I hope you do not mind me emailing you to ask for help about a mathematical problem whose answer is probably well known. You may know some algebraic geometer who can give a quick answer. My excuse is that the problem ties in with the beautiful demonstration in week229 of how the torus is a branched double cover of the sphere. You will have seen, I expect, the elegant proof in The Secret Blogging Seminar of Poncelet’s Porism. I raised the problem there, but have not found an answer. Being now retired and having little contact with academic life, I thought I might be impudent enough to see if you could ask your friends - I know you have a very busy life, but then if you do not already know the answer I am guessing you will want to find it.

The background to the problem is: $S$ and $T$ are nondegenerate conics having 4 distinct points of intersection (everything over $\mathbb{C}$, of course). $X$ is the variety of pairs $(s,t)$, $s$ a point of $S$, $t$ a tangent to $T$, such that $s$ lies on $t$. $S$ is topologically a sphere (projective line) and $X$ is a double cover of $S$ with branches over the 4 points of intersection of $S$ with $T$. So $X$ is a torus. It is therefore a model of the algebraic theory that is the affine part of the theory of abelian groups. This theory is generated by the 3-ary operation which, were we to choose a zero, we might write as

$$(x_1,x_2,x_3) \mapsto x_1-x_2+x_3.$$

It is not hard to see from the picture in week229 that this operation takes any 3 of the 4 points of intersection of $S$ with $T$ into the fourth. My question is: is there a geometric construction for this operation? I am looking for something old-fashioned; drawn with a stick in the sand, without vulgar reference to anything Poncelet would not have heard of. Given $x_1 = (s_1,t_1)$ and $x_3 = (s_3,t_3)$ one is looking for the involution on $X$ taking $x_2$ to $x_1-x_2+x_3$, described geometrically purely in terms of $s_1,s_2,s_3,t_1,t_2,t_3$ and no messing with algebra if possible. It has to be well known, but I stopped doing this stuff at school.

Best wishes

Gavin Wraith